Metrics of special curvature in differential geometry

这个项目属于微分几何领域，微分几何是数学的一个分支，主要研究光滑物体（称为流形）的长度和曲率特性，通常是在更高的维度上。微分几何是现代数学的重要组成部分，与代数、分析、拓扑学和理论物理等许多领域有着直接而密切的联系。微分几何研究的一个中心领域是黎曼流形：光滑物体被赋予黎曼度规，它允许人们测量长度和角度。黎曼流形的曲率特性被编码在黎曼曲率张量中。这以一种复杂和非线性的方式依赖于度规，因此曲率条件涉及非线性偏微分方程，通常难以分析。例子包括爱因斯坦方程和里奇孤子方程，它们都是微分几何中的关键方程，因为它们与拓扑和数学物理以及通过里奇流进行的几何分析有潜在的联系。黎曼流形上的其他重要方程也涉及辅助数据，如流形上向量或旋量束的连接或部分。这个项目将涉及使用对称假设，通过将曲率方程简化为低阶常微分方程或偏微分方程的系统来简化曲率方程。要考虑的方程包括爱因斯坦方程以及厄米流形或Kähler流形上的施特罗明格系统。爱因斯坦方程在黎曼几何中是至关重要的，然而我们对四维以上的解的理解相对较差，这是由于缺乏例子。施特罗明格系统起源于数学物理，通过研究扭转背景下的超弦，涉及到一个厄米度规和流形上的一个连接。尽管数学家和物理学家都对它有很好的动机和研究，但施特罗明格系统解的非平凡例子的缺乏再次意味着我们对该系统的理解存在明显的差距。该方法将包括使用齐次空间和李群理论来实现对称约简，以及使用非线性动力系统和椭圆偏微分方程的技术来分析所得到的系统。特别是，这种方法尚未被用于研究施特罗明格系统，因此代表了一个新的研究方向。该方法已被证明在其他相关情况下非常强大，因此可以希望在项目设置中使用该工具将同样富有成效。该项目将有助于更好地理解这些方程，包括解的新的存在性结果，某些情况下可能的显式解，以及唯一性和模性的结果。这些结果不仅与研究微分几何的纯数学工作者有关，也与数学物理学家有关。该项目属于EPSRC的研究领域几何和拓扑，但也与分析和数学物理有很强的联系。

This project is in the area of differential geometry, the branch of mathematics primarily concerned with the length and curvature properties of smooth objects (called manifolds), often in higher dimensions. Differential geometry is a key part of modern mathematics with direct and close links to numerous areas including algebra, analysis, topology and theoretical physics.A central area of study in differential geometry is Riemannian manifolds: smooth objects endowed with a Riemannian metric, which allows one to measure lengths and angles. The curvature properties of a Riemannian manifold are encoded in the Riemann curvature tensor. This depends in a complicated and nonlinear way on the metric, and so curvature conditions involve nonlinear partial differential equations which are usually difficult to analyse. Examples include the Einstein equations and the Ricci soliton equations, both of which are key equations in differential geometry because of their potential links to topology and mathematical physics, as well as to geometric analysis through the Ricci flow. Other important equations on Riemannian manifolds also involve auxiliary data such as connexions or sections of vector or spinor bundles on the manifold.This project will involve the use of symmetry assumptions to simplify such curvature equations by reducing them to systems of ordinary differential equations or partial differential equations of lower order. Equations to be considered include the Einstein equation and also the Strominger system on a Hermitian or Kähler manifold. The Einstein equation is of fundamental importance in Riemannian geometry, yet our understanding of solutions is relatively poor in dimensions above four, and this is due in poor to a lack of examples. The Strominger system originated in mathematical physics through the study of superstrings in torsion backgrounds, and involves a Hermitian metric and a connexion on the manifold. Although it is well-motivated and studied both by mathematicians and physicists, the paucity of non-trivial examples of solutions to the Strominger system again means that there is a clear gap in our understanding of the system. The methodology will involve using the theory of homogeneous spaces and Lie groups to implement the symmetry reduction, as well as using techniques from nonlinear dynamical systems and elliptic partial differential equations to analyse the resulting systems. This methodology, in particular, has not been used to study the Strominger system, and so represents a novel research direction. The methodology has proved to be very powerful in other related situations, and so one can hope that it will be similarly fruitful to utilize this tool in the setting of the project.The project should lead to a better understanding of these equations, including new existence results for solutions, possible explicit solutions in some cases, and results on uniqueness and moduli. The results will be of relevance not only to pure mathematicians working in differential geometry but also to mathematical physicists.This project falls within EPSRC research area Geometry and Topology, but also has strong links to Analysis and Mathematical Physics.